The spectral scheme for spacetime geometry is a totally new framework for quantitatively describing the spacetime geometry in terms of the spectra of a certain elliptic operator (typically the Laplacian operator) on the space in question. The central idea of the framework can be symbolically stated as ``Let us hear the shape of the Universe!" There are several advantages of this framework compared to the traditional geometrical description in terms of the Riemannian metric. After sketching the basics of the spectral scheme, we give a new formula for the Einstein-Hilbert action, which is a central quantity for the general relativity theory, in terms of the spectral scheme. We then pay attention to its application to the quantum universes and see how the quantum fluctuations of spacetimes can be effectively described in terms of the spectral scheme.
Key words: General relativity, spacetime structures, spectral scheme, cosmology.
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